Classical recursion theory
Volume II of Classical Recursion Theory describes the universe from a local (bottom-up
or synthetical) point of view, and covers the whole spectrum, from the
recursive to the arithmetical sets.
The first half of the book provides a detailed picture of the computable
sets from the perspective of Theoretical Computer Science. Besides giving a
detailed description of the theories of abstract Complexity Theory and of Inductive Inference, it contributes a uniform picture of the most basic complexity
classes, ranging from small time and space bounds to the elementary functions,
with a particular attention to polynomial time and space computability. It also
deals with primitive recursive functions and larger classes, which are of
interest to the proof theorist.
The second half of the book starts with the classical theory of recursively
enumerable sets and degrees, which constitutes the core of Recursion or
Computability Theory. Unlike other texts, usually confined to the Turing
degrees, the book covers a variety of other strong reducibilities, studying
both their individual structures and their mutual relationships. The last
chapters extend the theory to limit sets and arithmetical sets. The volume
ends with the first textbook treatment of the enumeration degrees, which
admit a number of applications from algebra to the Lambda Calculus.
The book is a valuable source of information for anyone interested in
Complexity and Computability Theory. The student will appreciate the detailed
but informal account of a wide variety of basic topics, while the specialist
will find a wealth of material sketched in exercises and asides. A massive
bibliography of more than a thousand titles completes the treatment on the
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THEORIES OF RECURSIVE FUNCTIONS
HIERARCHIES OF RECURSIVE FUNCTIONS
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1-generic Ambos-Spies analogue arithmetical automorphism bounded primitive recursion bounded quantifiers characterization class of sets closure co-NP coding coinfinite complexity classes computable function consider construction contains Conversely Corollary countable crossing sequences defined definition deterministic e-splitting elementarily equivalent elementary elements ensure Exercises exists exponential EXPSPACE first-order follows function g given Hartmanis hence Hierarchy Theorem Hint hyperhypersimple induction hypothesis infinite initial segment input isomorphic iteration Jockusch jump Lachlan Lemma length LOGSPACE m-degrees maximal minimal degree minimal pair NEXP nondeterministic Turing machine nontrivial notion NP sets obtained oracle ordinal otherwise partial recursive functions particular POLYEXP polynomial time computable polynomially bounded primitive recursive functions priority problem proof properties Proposition proved PSPACE r-maximal r.e. degrees r.e. set recursive enumeration recursive sets relativization requirements result satisfied Section sets acceptable sets computable Slaman smallest space stage strings subset suppose T-complete theory total recursive functions Turing degrees uniformly